Logarithms

To understand logarithms, you should first understand
exponents
because a logarithm is an exponent.

Finding a logarithm
(base
b
) of a number is like answering the question:

To what power do I have to raise
b
to get this number?

Or, in math notation:

log
b
a
=
x
means
b
x
=
a

Examples:

log
7
49
=
2
, since
7
2
=
49

log
2
32
=
5
, since
2
5
=
32

log
10
0.01
=
−
2
, since
10
−
2
=
0.01

This can be a little confusing. Remember that the base in the logarithm equation (the small subscripted number) is also the base in the power equation (the number raised to the power), and they stay on the left side.

In real life, we usually only use two bases:
log
10
, also called the
common logarithm
, and
log
e
, where
e
≈
2.71828
, also called the
natural logarithm
.

log
10
x
is usually written
log
x
, with the base understood to be
10
.
log
e
x
is written
ln
x
.